For now that's enough for use to deduce the pattern. If we consider s(N), the number of elements in the set S(N) we have

s(1) = 2s(2) = 3s(3) = 5s(4) = 8

This looks familiar, but before I tell you what it is, let us look at the generation of the sets in a bit more detail. First, let's turn the rules around: what the rules imply is that we can always add a 0 to a string, but we can only add a 1 if it ends in a 0. What this means is for the set S(N+1) is that we always have s(N) strings generated by the 0-rule. How about the 1-rule. To determine the number of 1's we can add, we need to know how many strings in S(N) end in 0. But that is easy: for in S(N) we added s(N-1) 0's.